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We consider one-sided weight classes of Muckenhoupt type, but larger than the classical Muckenhoupt classes, and study the boundedness of one-sided oscillatory integral operators on weighted Lebesgue spaces using interpolation of operators with change of measures.

Oscillatory integrals in one form or another have been an essential part of harmonic analysis from the very beginnings of that subject; three chapters are devoted to them in the celebrated Stein's book [

We state a celebrated result of Ricci and Stein on oscillatory integrals as follows.

Let

Weighted inequalities arise naturally in harmonic analysis, but their use is best justified by the variety of applications in which they appear. It is worth pointing out that many authors are interested in the inequalities when the weight functions belong to the Muckenhoupt classes ([

In 1992, Lu and Zhang [

Let

We point out that Theorems

The study of weights for one-sided operators was motivated not only by the generalization of the theory of both-sided ones, but also by their natural appearance in harmonic analysis; for example, they are required when we treat the one-sided Hardy-Littlewood maximal operator [

The smallest constant

Let

The one-sided weight classes are of interest, not only because they control the boundedness of the one-sided Hardy-Littlewood maximal operator, but also because they are the right classes for the weighted estimates of one-sided Calderón-Zygmund singular integral operators [

Let

Theorem

Highly inspired by the above statements for oscillatory singular integral operators and one-sided operator theory, in [

We first recall the definition of one-sided oscillatory integral operator as

Let

there exists constant

there exists constant

The rest of this paper is devoted to the argument for Theorem

Let

there exist

According to the definition of

Let

For

We say a weight

Let

A weight

Combining the results in [

Let

By Lemma

The inequality

Let us fix

To prove Theorem

Suppose that

Lemmas

In this section, we will prove Theorem

If

Let

We will now prove that the conclusion of Theorem

If

Suppose

Now, let

Write

We consider first the estimates for

Observe that if

Notice that if

Again observe that if

Combining (

Evidently, if

From (

We proceed with the proof of Theorem

From (

In this case, we write

(2) We omit the details, since they are very similar to that of the proof of (1) with

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was partially supported by NSF of China (Grant nos. 11271175, 10931001, and 11301249), NSF of Shandong Province (Grant no. ZR2012AQ026), the AMEP and DYSP of Linyi University, and the Key Laboratory of Mathematics and Complex System (Beijing Normal University), Ministry of Education, China.